sound of the water because he says he hears it

While I stand on the roadway, or on the pavements grey

In other words, he hears it while he is the midst of the city and a trafficked roadway. By saying he hears it in his "heart," he means he doesn't literally hear it; he can't really hear the sound because he is too far away from that peaceful setting. It is something he remembers or imagines that gives him comfort and peace amid the busy pace of life.

At the end of the poem, the speaker says that he hears the sound of the waters of Lake Innisfree lapping gently at the shores. He also states that he hears this sound when he is standing on a roadway or on a "pavement," which is a British term for a sidewalk, implying he is in a city.

If he is in a city or the edge of a busy road, how can he be hearing the calm, lapping waters of a beloved lake that is far away? He explains that he hears the sound in his "deep heart's core."

He means that he hears the sound in his imagination. However, it is significant that he does not say "imagination." The heart is associated with emotion. In the phrase "deep heart's core," the speaker is trying to communicate that he feels the memory of the soothing lapping of the waters as a calming emotion amid all the hectic, urban business of life. He carries his time with nature inside of himself, and it helps keep him at peace.

In the poem 'The Lake Isle of Inisfree' by William Butler Yeats, the poet is feeling homesick for the landscape he loves in the west of Ireland. Although he came from an anglo-irish background he developed a deep love for the Irish people and their cause (freedom) and culture - he worked to develop drama and theatre after the setting up of the Irish Free State. This love sprang from very deep in his psyche (his 'deep heart's core.') It is also reminiscent of the Irish Gaelic language which is very poetic and talks in terms of things belonging to other things. For example, another example would be 'my heart's pulse' which is a beautiful term for a child or a deeply loved person. So here Yeats is referring both to his own heart,the profundity of love and also to the centre of the earth itself. In a similar way he writes 'the centre cannot hold' and 'things fall apart' in his poems about chaos and revolution. Being 'centred' was very important to Yeats and to many Irish people today.

In the final stanza, the speaker says,

I will arise and go now, for always night and day
I hear lake water lapping with low sounds by the shore;
While I stand on the roadway, or on the pavements grey,
I hear it in the deep heart's core.

Thus, no matter where the speaker is—even if he is on the road somewhere far away or surrounded by gray pavement in a city—he can hear the lapping lake water and the sounds of the shore at Innisfree. When he says that he hears these sounds "in the deep heart's core," he seems to be talking about both the core (or very center) of his own heart as well as the core (or very center) of the world, for him. For the speaker, the Lake Isle of Innisfree is the perfect place, a place of "peace" and solitude and beautiful midnights that "glimmer" and noons that glow "purple." It feels vital and perfect and pure, and thus it is, figuratively, in the heart of the earth. The speaker can feel the place calling to him from within his own heart as well as through the earth around him, no matter where he is.

When Yeats chooses to use “in the deep heart’s core” instead of “in the core of my heart,” the speaker is distancing the “peace” he feels from himself. This reinforces the idea that the peace found in the simple life of nature is universal.

And I shall have some peace there, for peace
comes dropping slow,

By referring to the heart with “the” instead of “my” he also produces a double metaphor for the heart of the island and the speaker’s heart. In other words, by retreating to the peace of the island he is entering the “heart” of nature and his own heart.

Of course, “core” does rhyme with “shore” in the third line of the last stanza, so arranging the words this way does maintain the abab rhyme scheme and thus both the structure and internal logic of the poem.

Yeats's speaker dreams in this poem of going to an isle on Lake Innisfree in Ireland. He imagines he will build a cabin there, grow some beans, keep some bees, and experience a simple, idyllic, peaceful, and natural life away from the demands of civilization.

At the end of the poem, the speaker thinks about the quiet sound of the "lake water lapping" the shore at the lake isle of Innisfree:

While I stand on the roadway, or on the pavements [sidewalks] grey, I hear it [the sounds of Innisfree] in the deep heart’s core.